LMFDB - Embedded newform 1900.3.g.d.949.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(949,1900)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1900.949");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			949.7
Character	$\chi$	$=$	1900.949
Dual form			1900.3.g.d.949.8

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.93221 q^{3} -5.62705i q^{7} -0.402160 q^{9} +O(q^{10})$	$q-2.93221 q^{3} -5.62705i q^{7} -0.402160 q^{9} -17.4504 q^{11} -7.35569 q^{13} +25.3144i q^{17} +(2.59985 + 18.8213i) q^{19} +16.4997i q^{21} +19.3107i q^{23} +27.5691 q^{27} +4.89731i q^{29} -45.2147i q^{31} +51.1681 q^{33} -25.2986 q^{37} +21.5684 q^{39} +16.2721i q^{41} +28.0559i q^{43} +38.9299i q^{47} +17.3363 q^{49} -74.2270i q^{51} +81.9687 q^{53} +(-7.62329 - 55.1879i) q^{57} -61.0035i q^{59} -94.1024 q^{61} +2.26297i q^{63} -109.942 q^{67} -56.6231i q^{69} -57.6965i q^{71} +36.4469i q^{73} +98.1940i q^{77} -112.540i q^{79} -77.2188 q^{81} -42.1499i q^{83} -14.3599i q^{87} -101.024i q^{89} +41.3909i q^{91} +132.579i q^{93} -148.954 q^{97} +7.01784 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 104 q^{9}+O(q^{10}) $ 28 * q + 104 * q^9	$ 28 q + 104 q^{9} + 8 q^{11} + 58 q^{19} + 112 q^{39} - 276 q^{49} - 100 q^{61} + 132 q^{81} - 184 q^{99}+O(q^{100}) $ 28 * q + 104 * q^9 + 8 * q^11 + 58 * q^19 + 112 * q^39 - 276 * q^49 - 100 * q^61 + 132 * q^81 - 184 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.93221			−0.977402			−0.488701	−	0.872451i	$-0.662529\pi$
$3$	−2.93221			−0.977402			−0.488701	+	0.872451i	$0.662529\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	5.62705i		−	0.803864i	−0.915669	−	0.401932i	$-0.868339\pi$
$7$		−	5.62705i		−	0.803864i	0.915669	−	0.401932i	$-0.131661\pi$
$8$		0			0
$9$	−0.402160			−0.0446844
$10$		0			0
$11$	−17.4504			−1.58640			−0.793198	−	0.608963i	$-0.791586\pi$
$11$	−17.4504			−1.58640			−0.793198	+	0.608963i	$0.791586\pi$
$12$		0			0
$13$	−7.35569			−0.565823			−0.282911	−	0.959146i	$-0.591300\pi$
$13$	−7.35569			−0.565823			−0.282911	+	0.959146i	$0.591300\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			25.3144i			1.48908i	0.667577	+	0.744541i	$0.267332\pi$
$17$			25.3144i			1.48908i	−0.667577	+	0.744541i	$0.732668\pi$
$18$		0			0
$19$	2.59985	+	18.8213i	0.136834	+	0.990594i
$19$	2.59985	+	18.8213i	0.136834	+	0.990594i
$20$		0			0
$21$			16.4997i			0.785699i
$22$		0			0
$23$			19.3107i			0.839598i	0.907617	+	0.419799i	$0.137899\pi$
$23$			19.3107i			0.839598i	−0.907617	+	0.419799i	$0.862101\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	27.5691			1.02108
$28$		0			0
$29$			4.89731i			0.168873i	0.996429	+	0.0844363i	$0.0269089\pi$
$29$			4.89731i			0.168873i	−0.996429	+	0.0844363i	$0.973091\pi$
$30$		0			0
$31$		−	45.2147i		−	1.45854i	−0.684226	−	0.729270i	$-0.739860\pi$
$31$		−	45.2147i		−	1.45854i	0.684226	−	0.729270i	$-0.260140\pi$
$32$		0			0
$33$	51.1681			1.55055
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−25.2986			−0.683746			−0.341873	−	0.939746i	$-0.611061\pi$
$37$	−25.2986			−0.683746			−0.341873	+	0.939746i	$0.611061\pi$
$38$		0			0
$39$	21.5684			0.553036
$40$		0			0
$41$			16.2721i			0.396881i	0.980113	+	0.198440i	$0.0635876\pi$
$41$			16.2721i			0.396881i	−0.980113	+	0.198440i	$0.936412\pi$
$42$		0			0
$43$			28.0559i			0.652462i	0.945290	+	0.326231i	$0.105779\pi$
$43$			28.0559i			0.652462i	−0.945290	+	0.326231i	$0.894221\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			38.9299i			0.828296i	0.910209	+	0.414148i	$0.135920\pi$
$47$			38.9299i			0.828296i	−0.910209	+	0.414148i	$0.864080\pi$
$48$		0			0
$49$	17.3363			0.353802
$50$		0			0
$51$		−	74.2270i		−	1.45543i
$52$		0			0
$53$	81.9687			1.54658			0.773289	−	0.634053i	$-0.218610\pi$
$53$	81.9687			1.54658			0.773289	+	0.634053i	$0.218610\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−7.62329	−	55.1879i	−0.133742	−	0.968209i
$58$		0			0
$59$		−	61.0035i		−	1.03396i	−0.855998	−	0.516979i	$-0.827056\pi$
$59$		−	61.0035i		−	1.03396i	0.855998	−	0.516979i	$-0.172944\pi$
$60$		0			0
$61$	−94.1024			−1.54266			−0.771331	−	0.636434i	$-0.780409\pi$
$61$	−94.1024			−1.54266			−0.771331	+	0.636434i	$0.780409\pi$
$62$		0			0
$63$			2.26297i			0.0359202i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−109.942			−1.64092			−0.820459	−	0.571705i	$-0.806282\pi$
$67$	−109.942			−1.64092			−0.820459	+	0.571705i	$0.806282\pi$
$68$		0			0
$69$		−	56.6231i		−	0.820625i
$70$		0			0
$71$		−	57.6965i		−	0.812627i	−0.913734	−	0.406313i	$-0.866814\pi$
$71$		−	57.6965i		−	0.812627i	0.913734	−	0.406313i	$-0.133186\pi$
$72$		0			0
$73$			36.4469i			0.499272i	0.968340	+	0.249636i	$0.0803110\pi$
$73$			36.4469i			0.499272i	−0.968340	+	0.249636i	$0.919689\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			98.1940i			1.27525i
$78$		0			0
$79$		−	112.540i		−	1.42455i	−0.701900	−	0.712276i	$-0.747665\pi$
$79$		−	112.540i		−	1.42455i	0.701900	−	0.712276i	$-0.252335\pi$
$80$		0			0
$81$	−77.2188			−0.953319
$82$		0			0
$83$		−	42.1499i		−	0.507830i	−0.967227	−	0.253915i	$-0.918282\pi$
$83$		−	42.1499i		−	0.507830i	0.967227	−	0.253915i	$-0.0817184\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	14.3599i		−	0.165057i
$88$		0			0
$89$		−	101.024i		−	1.13510i	−0.823339	−	0.567549i	$-0.807892\pi$
$89$		−	101.024i		−	1.13510i	0.823339	−	0.567549i	$-0.192108\pi$
$90$		0			0
$91$			41.3909i			0.454845i
$92$		0			0
$93$			132.579i			1.42558i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−148.954			−1.53560			−0.767802	−	0.640687i	$-0.778650\pi$
$97$	−148.954			−1.53560			−0.767802	+	0.640687i	$0.778650\pi$
$98$		0			0
$99$	7.01784			0.0708872
$100$		0			0
$101$	35.3143			0.349647			0.174823	−	0.984600i	$-0.444065\pi$
$101$	35.3143			0.349647			0.174823	+	0.984600i	$0.444065\pi$
$102$		0			0
$103$	63.2704			0.614276			0.307138	−	0.951665i	$-0.400629\pi$
$103$	63.2704			0.614276			0.307138	+	0.951665i	$0.400629\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	197.887			1.84942			0.924708	−	0.380677i	$-0.124309\pi$
$107$	197.887			1.84942			0.924708	+	0.380677i	$0.124309\pi$
$108$		0			0
$109$			179.503i			1.64682i	0.567450	+	0.823408i	$0.307930\pi$
$109$			179.503i			1.64682i	−0.567450	+	0.823408i	$0.692070\pi$
$110$		0			0
$111$	74.1807			0.668295
$112$		0			0
$113$	20.3688			0.180255			0.0901276	−	0.995930i	$-0.471273\pi$
$113$	20.3688			0.180255			0.0901276	+	0.995930i	$0.471273\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.95817			0.0252835
$118$		0			0
$119$	142.445			1.19702
$120$		0			0
$121$	183.515			1.51665
$122$		0			0
$123$		−	47.7132i		−	0.387912i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−93.0448			−0.732636			−0.366318	−	0.930490i	$-0.619382\pi$
$127$	−93.0448			−0.732636			−0.366318	+	0.930490i	$0.619382\pi$
$128$		0			0
$129$		−	82.2656i		−	0.637718i
$130$		0			0
$131$	−207.262			−1.58215			−0.791077	−	0.611717i	$-0.790479\pi$
$131$	−207.262			−1.58215			−0.791077	+	0.611717i	$0.790479\pi$
$132$		0			0
$133$	105.908	−	14.6295i	0.796303	−	0.109996i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	143.376i		−	1.04654i	−0.852168	−	0.523268i	$-0.824713\pi$
$137$		−	143.376i		−	1.04654i	0.852168	−	0.523268i	$-0.175287\pi$
$138$		0			0
$139$	158.125			1.13759			0.568794	−	0.822480i	$-0.307410\pi$
$139$	158.125			1.13759			0.568794	+	0.822480i	$0.307410\pi$
$140$		0			0
$141$		−	114.151i		−	0.809579i
$142$		0			0
$143$	128.360			0.897619
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−50.8337			−0.345807
$148$		0			0
$149$	−211.739			−1.42107			−0.710534	−	0.703663i	$-0.751547\pi$
$149$	−211.739			−1.42107			−0.710534	+	0.703663i	$0.751547\pi$
$150$		0			0
$151$		−	242.067i		−	1.60310i	−0.597930	−	0.801548i	$-0.704010\pi$
$151$		−	242.067i		−	1.60310i	0.597930	−	0.801548i	$-0.295990\pi$
$152$		0			0
$153$		−	10.1804i		−	0.0665388i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			137.436i			0.875386i	0.899125	+	0.437693i	$0.144204\pi$
$157$			137.436i			0.875386i	−0.899125	+	0.437693i	$0.855796\pi$
$158$		0			0
$159$	−240.349			−1.51163
$160$		0			0
$161$	108.663			0.674923
$162$		0			0
$163$		−	173.770i		−	1.06607i	−0.846092	−	0.533037i	$-0.821051\pi$
$163$		−	173.770i		−	1.06607i	0.846092	−	0.533037i	$-0.178949\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.88276			−0.0352261			−0.0176131	−	0.999845i	$-0.505607\pi$
$167$	−5.88276			−0.0352261			−0.0176131	+	0.999845i	$0.505607\pi$
$168$		0			0
$169$	−114.894			−0.679845
$170$		0			0
$171$	−1.04555	−	7.56917i	−0.00611435	−	0.0442641i
$172$		0			0
$173$	254.130			1.46896			0.734480	−	0.678630i	$-0.237426\pi$
$173$	254.130			1.46896			0.734480	+	0.678630i	$0.237426\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			178.875i			1.01059i
$178$		0			0
$179$		−	144.808i		−	0.808984i	−0.914541	−	0.404492i	$-0.867448\pi$
$179$		−	144.808i		−	0.808984i	0.914541	−	0.404492i	$-0.132552\pi$
$180$		0			0
$181$		−	98.4770i		−	0.544072i	−0.962287	−	0.272036i	$-0.912303\pi$
$181$		−	98.4770i		−	0.544072i	0.962287	−	0.272036i	$-0.0876969\pi$
$182$		0			0
$183$	275.928			1.50780
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	441.745i		−	2.36227i
$188$		0			0
$189$		−	155.133i		−	0.820807i
$190$		0			0
$191$	326.880			1.71141			0.855706	−	0.517462i	$-0.173123\pi$
$191$	326.880			1.71141			0.855706	+	0.517462i	$0.173123\pi$
$192$		0			0
$193$	140.497			0.727965			0.363982	−	0.931406i	$-0.381417\pi$
$193$	140.497			0.727965			0.363982	+	0.931406i	$0.381417\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	344.527i		−	1.74887i	−0.485146	−	0.874433i	$-0.661233\pi$
$197$		−	344.527i		−	1.74887i	0.485146	−	0.874433i	$-0.338767\pi$
$198$		0			0
$199$	202.257			1.01637			0.508184	−	0.861249i	$-0.330317\pi$
$199$	202.257			1.01637			0.508184	+	0.861249i	$0.330317\pi$
$200$		0			0
$201$	322.371			1.60384
$202$		0			0
$203$	27.5574			0.135751
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	7.76601i		−	0.0375170i
$208$		0			0
$209$	−45.3683	−	328.438i	−0.217073	−	1.57147i
$210$		0			0
$211$			126.032i			0.597308i	0.954361	+	0.298654i	$0.0965377\pi$
$211$			126.032i			0.597308i	−0.954361	+	0.298654i	$0.903462\pi$
$212$		0			0
$213$			169.178i			0.794263i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−254.426			−1.17247
$218$		0			0
$219$		−	106.870i		−	0.487990i
$220$		0			0
$221$		−	186.205i		−	0.842556i
$222$		0			0
$223$	359.998			1.61434			0.807169	−	0.590320i	$-0.200998\pi$
$223$	359.998			1.61434			0.807169	+	0.590320i	$0.200998\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−82.6104			−0.363922			−0.181961	−	0.983306i	$-0.558245\pi$
$227$	−82.6104			−0.363922			−0.181961	+	0.983306i	$0.558245\pi$
$228$		0			0
$229$	42.1892			0.184232			0.0921161	−	0.995748i	$-0.470637\pi$
$229$	42.1892			0.184232			0.0921161	+	0.995748i	$0.470637\pi$
$230$		0			0
$231$		−	287.925i		−	1.24643i
$232$		0			0
$233$			125.229i			0.537463i	0.963215	+	0.268731i	$0.0866044\pi$
$233$			125.229i			0.537463i	−0.963215	+	0.268731i	$0.913396\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			329.989i			1.39236i
$238$		0			0
$239$	177.271			0.741720			0.370860	−	0.928689i	$-0.379063\pi$
$239$	177.271			0.741720			0.370860	+	0.928689i	$0.379063\pi$
$240$		0			0
$241$			86.2189i			0.357755i	0.983871	+	0.178877i	$0.0572465\pi$
$241$			86.2189i			0.357755i	−0.983871	+	0.178877i	$0.942753\pi$
$242$		0			0
$243$	−21.7001			−0.0893009
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−19.1237	−	138.444i	−0.0774238	−	0.560500i
$248$		0			0
$249$			123.592i			0.496354i
$250$		0			0
$251$	−300.306			−1.19644			−0.598219	−	0.801332i	$-0.704125\pi$
$251$	−300.306			−1.19644			−0.598219	+	0.801332i	$0.704125\pi$
$252$		0			0
$253$		−	336.980i		−	1.33194i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	232.749			0.905639			0.452820	−	0.891602i	$-0.350418\pi$
$257$	232.749			0.905639			0.452820	+	0.891602i	$0.350418\pi$
$258$		0			0
$259$			142.356i			0.549639i
$260$		0			0
$261$		−	1.96950i		−	0.00754598i
$262$		0			0
$263$		−	280.549i		−	1.06673i	−0.845886	−	0.533363i	$-0.820928\pi$
$263$		−	280.549i		−	1.06673i	0.845886	−	0.533363i	$-0.179072\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			296.223i			1.10945i
$268$		0			0
$269$			112.923i			0.419787i	0.977724	+	0.209893i	$0.0673117\pi$
$269$			112.923i			0.419787i	−0.977724	+	0.209893i	$0.932688\pi$
$270$		0			0
$271$	−95.7071			−0.353163			−0.176581	−	0.984286i	$-0.556504\pi$
$271$	−95.7071			−0.353163			−0.176581	+	0.984286i	$0.556504\pi$
$272$		0			0
$273$		−	121.367i		−	0.444566i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	53.9033i		−	0.194597i	−0.995255	−	0.0972983i	$-0.968980\pi$
$277$		−	53.9033i		−	0.194597i	0.995255	−	0.0972983i	$-0.0310201\pi$
$278$		0			0
$279$			18.1836i			0.0651740i
$280$		0			0
$281$		−	68.4137i		−	0.243465i	−0.992563	−	0.121732i	$-0.961155\pi$
$281$		−	68.4137i		−	0.243465i	0.992563	−	0.121732i	$-0.0388450\pi$
$282$		0			0
$283$			378.615i			1.33786i	0.743324	+	0.668932i	$0.233248\pi$
$283$			378.615i			1.33786i	−0.743324	+	0.668932i	$0.766752\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	91.5640			0.319038
$288$		0			0
$289$	−351.818			−1.21736
$290$		0			0
$291$	436.763			1.50090
$292$		0			0
$293$	−20.3415			−0.0694249			−0.0347125	−	0.999397i	$-0.511052\pi$
$293$	−20.3415			−0.0694249			−0.0347125	+	0.999397i	$0.511052\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−481.090			−1.61983
$298$		0			0
$299$		−	142.044i		−	0.475063i
$300$		0			0
$301$	157.872			0.524491
$302$		0			0
$303$	−103.549			−0.341746
$304$		0			0
$305$		0			0
$306$		0			0
$307$	315.163			1.02659			0.513296	−	0.858212i	$-0.328424\pi$
$307$	315.163			1.02659			0.513296	+	0.858212i	$0.328424\pi$
$308$		0			0
$309$	−185.522			−0.600394
$310$		0			0
$311$	188.579			0.606363			0.303182	−	0.952933i	$-0.401951\pi$
$311$	188.579			0.606363			0.303182	+	0.952933i	$0.401951\pi$
$312$		0			0
$313$			154.013i			0.492055i	0.969263	+	0.246028i	$0.0791254\pi$
$313$			154.013i			0.492055i	−0.969263	+	0.246028i	$0.920875\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.01072			−0.0126521			−0.00632606	−	0.999980i	$-0.502014\pi$
$317$	−4.01072			−0.0126521			−0.00632606	+	0.999980i	$0.502014\pi$
$318$		0			0
$319$		−	85.4598i		−	0.267899i
$320$		0			0
$321$	−580.247			−1.80762
$322$		0			0
$323$	−476.449	+	65.8135i	−1.47508	+	0.203757i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	526.340i		−	1.60960i
$328$		0			0
$329$	219.061			0.665838
$330$		0			0
$331$			97.1158i			0.293401i	0.989181	+	0.146701i	$0.0468654\pi$
$331$			97.1158i			0.293401i	−0.989181	+	0.146701i	$0.953135\pi$
$332$		0			0
$333$	10.1741			0.0305528
$334$		0			0
$335$		0			0
$336$		0			0
$337$	193.889			0.575337			0.287669	−	0.957730i	$-0.407120\pi$
$337$	193.889			0.575337			0.287669	+	0.957730i	$0.407120\pi$
$338$		0			0
$339$	−59.7257			−0.176182
$340$		0			0
$341$			789.014i			2.31382i
$342$		0			0
$343$		−	373.278i		−	1.08827i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			238.983i			0.688712i	0.938839	+	0.344356i	$0.111903\pi$
$347$			238.983i			0.688712i	−0.938839	+	0.344356i	$0.888097\pi$
$348$		0			0
$349$	46.7033			0.133820			0.0669102	−	0.997759i	$-0.478686\pi$
$349$	46.7033			0.133820			0.0669102	+	0.997759i	$0.478686\pi$
$350$		0			0
$351$	−202.790			−0.577749
$352$		0			0
$353$			323.922i			0.917627i	0.888533	+	0.458813i	$0.151725\pi$
$353$			323.922i			0.917627i	−0.888533	+	0.458813i	$0.848275\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−417.679			−1.16997
$358$		0			0
$359$	290.600			0.809472			0.404736	−	0.914434i	$-0.367363\pi$
$359$	290.600			0.809472			0.404736	+	0.914434i	$0.367363\pi$
$360$		0			0
$361$	−347.482	+	97.8649i	−0.962553	+	0.271094i
$362$		0			0
$363$	−538.104			−1.48238
$364$		0			0
$365$		0			0
$366$		0			0
$367$			218.143i			0.594396i	0.954816	+	0.297198i	$0.0960521\pi$
$367$			218.143i			0.594396i	−0.954816	+	0.297198i	$0.903948\pi$
$368$		0			0
$369$		−	6.54399i		−	0.0177344i
$370$		0			0
$371$		−	461.242i		−	1.24324i
$372$		0			0
$373$	−244.913			−0.656603			−0.328302	−	0.944573i	$-0.606476\pi$
$373$	−244.913			−0.656603			−0.328302	+	0.944573i	$0.606476\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	36.0231i		−	0.0955519i
$378$		0			0
$379$		−	550.720i		−	1.45309i	−0.687120	−	0.726543i	$-0.741126\pi$
$379$		−	550.720i		−	1.45309i	0.687120	−	0.726543i	$-0.258874\pi$
$380$		0			0
$381$	272.827			0.716080
$382$		0			0
$383$	349.640			0.912897			0.456449	−	0.889750i	$-0.349121\pi$
$383$	349.640			0.912897			0.456449	+	0.889750i	$0.349121\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	11.2829i		−	0.0291549i
$388$		0			0
$389$	348.489			0.895858			0.447929	−	0.894069i	$-0.352162\pi$
$389$	348.489			0.895858			0.447929	+	0.894069i	$0.352162\pi$
$390$		0			0
$391$	−488.840			−1.25023
$392$		0			0
$393$	607.736			1.54640
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	25.4266i		−	0.0640468i	−0.999487	−	0.0320234i	$-0.989805\pi$
$397$		−	25.4266i		−	0.0640468i	0.999487	−	0.0320234i	$-0.0101951\pi$
$398$		0			0
$399$	−310.545	+	42.8966i	−0.778309	+	0.107510i
$400$		0			0
$401$		−	638.917i		−	1.59331i	−0.604435	−	0.796654i	$-0.706601\pi$
$401$		−	638.917i		−	1.59331i	0.604435	−	0.796654i	$-0.293399\pi$
$402$		0			0
$403$			332.586i			0.825275i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	441.469			1.08469
$408$		0			0
$409$			522.383i			1.27722i	0.769531	+	0.638610i	$0.220490\pi$
$409$			522.383i			1.27722i	−0.769531	+	0.638610i	$0.779510\pi$
$410$		0			0
$411$			420.407i			1.02289i
$412$		0			0
$413$	−343.270			−0.831162
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−463.655			−1.11188
$418$		0			0
$419$	621.068			1.48226			0.741132	−	0.671360i	$-0.234289\pi$
$419$	621.068			1.48226			0.741132	+	0.671360i	$0.234289\pi$
$420$		0			0
$421$			303.897i			0.721845i	0.932596	+	0.360922i	$0.117538\pi$
$421$			303.897i			0.721845i	−0.932596	+	0.360922i	$0.882462\pi$
$422$		0			0
$423$		−	15.6561i		−	0.0370120i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			529.519i			1.24009i
$428$		0			0
$429$	−376.377			−0.877335
$430$		0			0
$431$			86.3824i			0.200423i	0.994966	+	0.100212i	$0.0319520\pi$
$431$			86.3824i			0.200423i	−0.994966	+	0.100212i	$0.968048\pi$
$432$		0			0
$433$	51.3805			0.118662			0.0593308	−	0.998238i	$-0.481103\pi$
$433$	51.3805			0.118662			0.0593308	+	0.998238i	$0.481103\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−363.453	+	50.2050i	−0.831701	+	0.114886i
$438$		0			0
$439$		−	736.743i		−	1.67823i	−0.543954	−	0.839115i	$-0.683074\pi$
$439$		−	736.743i		−	1.67823i	0.543954	−	0.839115i	$-0.316926\pi$
$440$		0			0
$441$	−6.97197			−0.0158095
$442$		0			0
$443$		−	50.2064i		−	0.113333i	−0.998393	−	0.0566664i	$-0.981953\pi$
$443$		−	50.2064i		−	0.113333i	0.998393	−	0.0566664i	$-0.0180471\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	620.863			1.38896
$448$		0			0
$449$			211.982i			0.472121i	0.971738	+	0.236060i	$0.0758563\pi$
$449$			211.982i			0.472121i	−0.971738	+	0.236060i	$0.924144\pi$
$450$		0			0
$451$		−	283.954i		−	0.629610i
$452$		0			0
$453$			709.792i			1.56687i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			438.611i			0.959761i	0.877334	+	0.479880i	$0.159320\pi$
$457$			438.611i			0.959761i	−0.877334	+	0.479880i	$0.840680\pi$
$458$		0			0
$459$			697.895i			1.52047i
$460$		0			0
$461$	−206.072			−0.447011			−0.223506	−	0.974703i	$-0.571750\pi$
$461$	−206.072			−0.447011			−0.223506	+	0.974703i	$0.571750\pi$
$462$		0			0
$463$		−	164.050i		−	0.354319i	−0.984182	−	0.177159i	$-0.943309\pi$
$463$		−	164.050i		−	0.354319i	0.984182	−	0.177159i	$-0.0566908\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			296.903i			0.635766i	0.948130	+	0.317883i	$0.102972\pi$
$467$			296.903i			0.635766i	−0.948130	+	0.317883i	$0.897028\pi$
$468$		0			0
$469$			618.646i			1.31908i
$470$		0			0
$471$		−	402.990i		−	0.855604i
$472$		0			0
$473$		−	489.585i		−	1.03506i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−32.9645			−0.0691080
$478$		0			0
$479$	−319.593			−0.667208			−0.333604	−	0.942713i	$-0.608265\pi$
$479$	−319.593			−0.667208			−0.333604	+	0.942713i	$0.608265\pi$
$480$		0			0
$481$	186.089			0.386879
$482$		0			0
$483$	−318.621			−0.659671
$484$		0			0
$485$		0			0
$486$		0			0
$487$	257.490			0.528727			0.264364	−	0.964423i	$-0.414838\pi$
$487$	257.490			0.528727			0.264364	+	0.964423i	$0.414838\pi$
$488$		0			0
$489$			509.530i			1.04198i
$490$		0			0
$491$	−897.982			−1.82888			−0.914442	−	0.404716i	$-0.867370\pi$
$491$	−897.982			−1.82888			−0.914442	+	0.404716i	$0.867370\pi$
$492$		0			0
$493$	−123.972			−0.251465
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−324.661			−0.653241
$498$		0			0
$499$	786.631			1.57641			0.788207	−	0.615410i	$-0.211010\pi$
$499$	786.631			1.57641			0.788207	+	0.615410i	$0.211010\pi$
$500$		0			0
$501$	17.2495			0.0344301
$502$		0			0
$503$			48.2524i			0.0959292i	0.998849	+	0.0479646i	$0.0152735\pi$
$503$			48.2524i			0.0959292i	−0.998849	+	0.0479646i	$0.984727\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	336.892			0.664482
$508$		0			0
$509$			974.111i			1.91377i	0.290460	+	0.956887i	$0.406192\pi$
$509$			974.111i			1.91377i	−0.290460	+	0.956887i	$0.593808\pi$
$510$		0			0
$511$	205.088			0.401347
$512$		0			0
$513$	71.6754	+	518.886i	0.139718	+	1.01147i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	679.341i		−	1.31401i
$518$		0			0
$519$	−745.162			−1.43576
$520$		0			0
$521$			442.872i			0.850043i	0.905183	+	0.425021i	$0.139733\pi$
$521$			442.872i			0.850043i	−0.905183	+	0.425021i	$0.860267\pi$
$522$		0			0
$523$	104.045			0.198940			0.0994699	−	0.995041i	$-0.468285\pi$
$523$	104.045			0.198940			0.0994699	+	0.995041i	$0.468285\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1144.58			2.17189
$528$		0			0
$529$	156.095			0.295076
$530$		0			0
$531$			24.5332i			0.0462018i
$532$		0			0
$533$		−	119.693i		−	0.224564i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			424.608i			0.790703i
$538$		0			0
$539$	−302.525			−0.561271
$540$		0			0
$541$	−463.659			−0.857042			−0.428521	−	0.903532i	$-0.640965\pi$
$541$	−463.659			−0.857042			−0.428521	+	0.903532i	$0.640965\pi$
$542$		0			0
$543$			288.755i			0.531777i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	456.359			0.834294			0.417147	−	0.908839i	$-0.363030\pi$
$547$	456.359			0.834294			0.417147	+	0.908839i	$0.363030\pi$
$548$		0			0
$549$	37.8442			0.0689330
$550$		0			0
$551$	−92.1736	+	12.7322i	−0.167284	+	0.0231075i
$552$		0			0
$553$	−633.266			−1.14515
$554$		0			0
$555$		0			0
$556$		0			0
$557$			188.836i			0.339024i	0.985528	+	0.169512i	$0.0542192\pi$
$557$			188.836i			0.339024i	−0.985528	+	0.169512i	$0.945781\pi$
$558$		0			0
$559$		−	206.370i		−	0.369178i
$560$		0			0
$561$			1295.29i			2.30889i
$562$		0			0
$563$	−561.830			−0.997923			−0.498961	−	0.866624i	$-0.666285\pi$
$563$	−561.830			−0.997923			−0.498961	+	0.866624i	$0.666285\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			434.514i			0.766339i
$568$		0			0
$569$			34.7296i			0.0610362i	0.999534	+	0.0305181i	$0.00971573\pi$
$569$			34.7296i			0.0610362i	−0.999534	+	0.0305181i	$0.990284\pi$
$570$		0			0
$571$	−215.723			−0.377798			−0.188899	−	0.981997i	$-0.560492\pi$
$571$	−215.723			−0.377798			−0.188899	+	0.981997i	$0.560492\pi$
$572$		0			0
$573$	−958.479			−1.67274
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	698.445i		−	1.21048i	−0.796044	−	0.605239i	$-0.793078\pi$
$577$		−	698.445i		−	1.21048i	0.796044	−	0.605239i	$-0.206922\pi$
$578$		0			0
$579$	−411.967			−0.711515
$580$		0			0
$581$	−237.180			−0.408226
$582$		0			0
$583$	−1430.38			−2.45349
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	312.489i		−	0.532349i	−0.963925	−	0.266175i	$-0.914240\pi$
$587$		−	312.489i		−	0.532349i	0.963925	−	0.266175i	$-0.0857597\pi$
$588$		0			0
$589$	851.000	−	117.551i	1.44482	−	0.199578i
$590$		0			0
$591$			1010.22i			1.70935i
$592$		0			0
$593$		−	1086.81i		−	1.83273i	−0.400339	−	0.916367i	$-0.631108\pi$
$593$		−	1086.81i		−	1.83273i	0.400339	−	0.916367i	$-0.368892\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−593.060			−0.993400
$598$		0			0
$599$			791.665i			1.32164i	0.750543	+	0.660822i	$0.229792\pi$
$599$			791.665i			1.32164i	−0.750543	+	0.660822i	$0.770208\pi$
$600$		0			0
$601$			113.205i			0.188362i	0.995555	+	0.0941808i	$0.0300232\pi$
$601$			113.205i			0.188362i	−0.995555	+	0.0941808i	$0.969977\pi$
$602$		0			0
$603$	44.2141			0.0733235
$604$		0			0
$605$		0			0
$606$		0			0
$607$	193.212			0.318306			0.159153	−	0.987254i	$-0.449124\pi$
$607$	193.212			0.318306			0.159153	+	0.987254i	$0.449124\pi$
$608$		0			0
$609$	−80.8040			−0.132683
$610$		0			0
$611$		−	286.357i		−	0.468669i
$612$		0			0
$613$		−	133.846i		−	0.218346i	−0.994023	−	0.109173i	$-0.965180\pi$
$613$		−	133.846i		−	0.218346i	0.994023	−	0.109173i	$-0.0348203\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	602.743i		−	0.976893i	−0.872594	−	0.488447i	$-0.837564\pi$
$617$		−	602.743i		−	0.976893i	0.872594	−	0.488447i	$-0.162436\pi$
$618$		0			0
$619$	−136.718			−0.220869			−0.110434	−	0.993883i	$-0.535224\pi$
$619$	−136.718			−0.220869			−0.110434	+	0.993883i	$0.535224\pi$
$620$		0			0
$621$			532.380i			0.857294i
$622$		0			0
$623$	−568.466			−0.912465
$624$		0			0
$625$		0			0
$626$		0			0
$627$	133.029	+	963.049i	0.212168	+	1.53596i
$628$		0			0
$629$		−	640.418i		−	1.01815i
$630$		0			0
$631$	1197.46			1.89772			0.948862	−	0.315691i	$-0.102236\pi$
$631$	1197.46			1.89772			0.948862	+	0.315691i	$0.102236\pi$
$632$		0			0
$633$		−	369.552i		−	0.583811i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−127.521			−0.200189
$638$		0			0
$639$			23.2032i			0.0363118i
$640$		0			0
$641$		−	144.826i		−	0.225938i	−0.993599	−	0.112969i	$-0.963964\pi$
$641$		−	144.826i		−	0.225938i	0.993599	−	0.112969i	$-0.0360360\pi$
$642$		0			0
$643$		−	423.852i		−	0.659179i	−0.944124	−	0.329589i	$-0.893090\pi$
$643$		−	423.852i		−	0.659179i	0.944124	−	0.329589i	$-0.106910\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	767.957i		−	1.18695i	−0.804852	−	0.593475i	$-0.797755\pi$
$647$		−	767.957i		−	1.18695i	0.804852	−	0.593475i	$-0.202245\pi$
$648$		0			0
$649$			1064.53i			1.64027i
$650$		0			0
$651$	746.029			1.14597
$652$		0			0
$653$		−	148.725i		−	0.227756i	−0.993495	−	0.113878i	$-0.963673\pi$
$653$		−	148.725i		−	0.227756i	0.993495	−	0.113878i	$-0.0363273\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	14.6575i		−	0.0223097i
$658$		0			0
$659$		−	1058.23i		−	1.60581i	−0.596109	−	0.802904i	$-0.703287\pi$
$659$		−	1058.23i		−	1.60581i	0.596109	−	0.802904i	$-0.296713\pi$
$660$		0			0
$661$			714.976i			1.08166i	0.841132	+	0.540829i	$0.181889\pi$
$661$			714.976i			1.08166i	−0.841132	+	0.540829i	$0.818111\pi$
$662$		0			0
$663$			545.991i			0.823516i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−94.5706			−0.141785
$668$		0			0
$669$	−1055.59			−1.57786
$670$		0			0
$671$	1642.12			2.44727
$672$		0			0
$673$	−624.276			−0.927602			−0.463801	−	0.885939i	$-0.653515\pi$
$673$	−624.276			−0.927602			−0.463801	+	0.885939i	$0.653515\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	79.4837			0.117406			0.0587029	−	0.998275i	$-0.481304\pi$
$677$	79.4837			0.117406			0.0587029	+	0.998275i	$0.481304\pi$
$678$		0			0
$679$			838.170i			1.23442i
$680$		0			0
$681$	242.231			0.355699
$682$		0			0
$683$	−918.987			−1.34552			−0.672758	−	0.739863i	$-0.734890\pi$
$683$	−918.987			−1.34552			−0.672758	+	0.739863i	$0.734890\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−123.707			−0.180069
$688$		0			0
$689$	−602.936			−0.875089
$690$		0			0
$691$	−555.126			−0.803366			−0.401683	−	0.915779i	$-0.631575\pi$
$691$	−555.126			−0.803366			−0.401683	+	0.915779i	$0.631575\pi$
$692$		0			0
$693$		−	39.4897i		−	0.0569837i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−411.919			−0.590988
$698$		0			0
$699$		−	367.197i		−	0.525318i
$700$		0			0
$701$	73.5776			0.104961			0.0524805	−	0.998622i	$-0.483287\pi$
$701$	73.5776			0.104961			0.0524805	+	0.998622i	$0.483287\pi$
$702$		0			0
$703$	−65.7724	−	476.152i	−0.0935596	−	0.677314i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	198.715i		−	0.281069i
$708$		0			0
$709$	578.030			0.815275			0.407638	−	0.913144i	$-0.366353\pi$
$709$	578.030			0.815275			0.407638	+	0.913144i	$0.366353\pi$
$710$		0			0
$711$			45.2589i			0.0636553i
$712$		0			0
$713$	873.131			1.22459
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−519.795			−0.724959
$718$		0			0
$719$	870.924			1.21130			0.605649	−	0.795732i	$-0.292913\pi$
$719$	870.924			1.21130			0.605649	+	0.795732i	$0.292913\pi$
$720$		0			0
$721$		−	356.026i		−	0.493794i
$722$		0			0
$723$		−	252.812i		−	0.349670i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	651.673i		−	0.896386i	−0.893937	−	0.448193i	$-0.852068\pi$
$727$		−	651.673i		−	0.896386i	0.893937	−	0.448193i	$-0.147932\pi$
$728$		0			0
$729$	758.599			1.04060
$730$		0			0
$731$	−710.217			−0.971569
$732$		0			0
$733$		−	16.2720i		−	0.0221992i	−0.999938	−	0.0110996i	$-0.996467\pi$
$733$		−	16.2720i		−	0.0221992i	0.999938	−	0.0110996i	$-0.00353318\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1918.52			2.60315
$738$		0			0
$739$	−970.120			−1.31275			−0.656373	−	0.754436i	$-0.727910\pi$
$739$	−970.120			−1.31275			−0.656373	+	0.754436i	$0.727910\pi$
$740$		0			0
$741$	56.0746	+	405.945i	0.0756742	+	0.547835i
$742$		0			0
$743$	−669.599			−0.901211			−0.450605	−	0.892723i	$-0.648792\pi$
$743$	−669.599			−0.901211			−0.450605	+	0.892723i	$0.648792\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			16.9510i			0.0226921i
$748$		0			0
$749$		−	1113.52i		−	1.48668i
$750$		0			0
$751$		−	798.869i		−	1.06374i	−0.846826	−	0.531870i	$-0.821489\pi$
$751$		−	798.869i		−	1.06374i	0.846826	−	0.531870i	$-0.178511\pi$
$752$		0			0
$753$	880.560			1.16940
$754$		0			0
$755$		0			0
$756$		0			0
$757$			843.963i			1.11488i	0.830218	+	0.557439i	$0.188216\pi$
$757$			843.963i			1.11488i	−0.830218	+	0.557439i	$0.811784\pi$
$758$		0			0
$759$			988.094i			1.30184i
$760$		0			0
$761$	174.165			0.228863			0.114431	−	0.993431i	$-0.463495\pi$
$761$	174.165			0.228863			0.114431	+	0.993431i	$0.463495\pi$
$762$		0			0
$763$	1010.07			1.32382
$764$		0			0
$765$		0			0
$766$		0			0
$767$			448.723i			0.585037i
$768$		0			0
$769$	−215.531			−0.280274			−0.140137	−	0.990132i	$-0.544754\pi$
$769$	−215.531			−0.280274			−0.140137	+	0.990132i	$0.544754\pi$
$770$		0			0
$771$	−682.469			−0.885174
$772$		0			0
$773$	−589.413			−0.762501			−0.381251	−	0.924472i	$-0.624507\pi$
$773$	−589.413			−0.762501			−0.381251	+	0.924472i	$0.624507\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	417.418i		−	0.537218i
$778$		0			0
$779$	−306.262	+	42.3050i	−0.393148	+	0.0543068i
$780$		0			0
$781$			1006.82i			1.28915i
$782$		0			0
$783$			135.014i			0.172432i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1145.89			−1.45602			−0.728009	−	0.685568i	$-0.759554\pi$
$787$	−1145.89			−1.45602			−0.728009	+	0.685568i	$0.759554\pi$
$788$		0			0
$789$			822.628i			1.04262i
$790$		0			0
$791$		−	114.616i		−	0.144901i
$792$		0			0
$793$	692.188			0.872873
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1240.74			1.55677			0.778384	−	0.627789i	$-0.216040\pi$
$797$	1240.74			1.55677			0.778384	+	0.627789i	$0.216040\pi$
$798$		0			0
$799$	−985.488			−1.23340
$800$		0			0
$801$			40.6277i			0.0507212i
$802$		0			0
$803$		−	636.011i		−	0.792044i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	331.113i		−	0.410301i
$808$		0			0
$809$	−214.567			−0.265225			−0.132612	−	0.991168i	$-0.542337\pi$
$809$	−214.567			−0.265225			−0.132612	+	0.991168i	$0.542337\pi$
$810$		0			0
$811$		−	443.871i		−	0.547313i	−0.961827	−	0.273657i	$-0.911767\pi$
$811$		−	443.871i		−	0.547313i	0.961827	−	0.273657i	$-0.0882332\pi$
$812$		0			0
$813$	280.633			0.345182
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−528.047	+	72.9409i	−0.646325	+	0.0892790i
$818$		0			0
$819$		−	16.6457i		−	0.0203245i
$820$		0			0
$821$	239.094			0.291223			0.145611	−	0.989342i	$-0.453485\pi$
$821$	239.094			0.291223			0.145611	+	0.989342i	$0.453485\pi$
$822$		0			0
$823$		−	367.550i		−	0.446598i	−0.974750	−	0.223299i	$-0.928317\pi$
$823$		−	367.550i		−	0.446598i	0.974750	−	0.223299i	$-0.0716826\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	432.940			0.523507			0.261753	−	0.965135i	$-0.415699\pi$
$827$	432.940			0.523507			0.261753	+	0.965135i	$0.415699\pi$
$828$		0			0
$829$			424.324i			0.511851i	0.966697	+	0.255925i	$0.0823801\pi$
$829$			424.324i			0.511851i	−0.966697	+	0.255925i	$0.917620\pi$
$830$		0			0
$831$			158.056i			0.190199i
$832$		0			0
$833$			438.858i			0.526841i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	1246.53i		−	1.48928i
$838$		0			0
$839$		−	1198.38i		−	1.42834i	−0.699974	−	0.714169i	$-0.746805\pi$
$839$		−	1198.38i		−	1.42834i	0.699974	−	0.714169i	$-0.253195\pi$
$840$		0			0
$841$	817.016			0.971482
$842$		0			0
$843$			200.603i			0.237963i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	1032.65i		−	1.21918i
$848$		0			0
$849$		−	1110.18i		−	1.30763i
$850$		0			0
$851$		−	488.535i		−	0.574071i
$852$		0			0
$853$		−	1631.88i		−	1.91311i	−0.291553	−	0.956555i	$-0.594172\pi$
$853$		−	1631.88i		−	1.91311i	0.291553	−	0.956555i	$-0.405828\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1704.86			−1.98933			−0.994666	−	0.103147i	$-0.967109\pi$
$857$	−1704.86			−1.98933			−0.994666	+	0.103147i	$0.967109\pi$
$858$		0			0
$859$	1190.68			1.38612			0.693061	−	0.720879i	$-0.256262\pi$
$859$	1190.68			1.38612			0.693061	+	0.720879i	$0.256262\pi$
$860$		0			0
$861$	−268.485			−0.311829
$862$		0			0
$863$	−1219.71			−1.41334			−0.706670	−	0.707543i	$-0.749803\pi$
$863$	−1219.71			−1.41334			−0.706670	+	0.707543i	$0.749803\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1031.60			1.18986
$868$		0			0
$869$			1963.86i			2.25990i
$870$		0			0
$871$	808.696			0.928468
$872$		0			0
$873$	59.9032			0.0686176
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1096.08			1.24981			0.624904	−	0.780702i	$-0.285138\pi$
$877$	1096.08			1.24981			0.624904	+	0.780702i	$0.285138\pi$
$878$		0			0
$879$	59.6455			0.0678561
$880$		0			0
$881$	−175.525			−0.199234			−0.0996170	−	0.995026i	$-0.531762\pi$
$881$	−175.525			−0.199234			−0.0996170	+	0.995026i	$0.531762\pi$
$882$		0			0
$883$			1733.12i			1.96277i	0.192055	+	0.981384i	$0.438485\pi$
$883$			1733.12i			1.96277i	−0.192055	+	0.981384i	$0.561515\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	134.522			0.151660			0.0758298	−	0.997121i	$-0.475839\pi$
$887$	134.522			0.151660			0.0758298	+	0.997121i	$0.475839\pi$
$888$		0			0
$889$			523.568i			0.588940i
$890$		0			0
$891$	1347.50			1.51234
$892$		0			0
$893$	−732.711	+	101.212i	−0.820505	+	0.113339i
$894$		0			0
$895$		0			0
$896$		0			0
$897$			416.502i			0.464328i
$898$		0			0
$899$	221.430			0.246307
$900$		0			0
$901$			2074.99i			2.30298i
$902$		0			0
$903$	−462.913			−0.512638
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−293.274			−0.323345			−0.161672	−	0.986844i	$-0.551689\pi$
$907$	−293.274			−0.323345			−0.161672	+	0.986844i	$0.551689\pi$
$908$		0			0
$909$	−14.2020			−0.0156238
$910$		0			0
$911$		−	303.681i		−	0.333349i	−0.986012	−	0.166674i	$-0.946697\pi$
$911$		−	303.681i		−	0.333349i	0.986012	−	0.166674i	$-0.0533029\pi$
$912$		0			0
$913$			735.531i			0.805620i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1166.27i			1.27184i
$918$		0			0
$919$	−1466.07			−1.59529			−0.797644	−	0.603128i	$-0.793921\pi$
$919$	−1466.07			−1.59529			−0.797644	+	0.603128i	$0.793921\pi$
$920$		0			0
$921$	−924.125			−1.00339
$922$		0			0
$923$			424.398i			0.459803i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−25.4448			−0.0274486
$928$		0			0
$929$	1278.74			1.37647			0.688234	−	0.725489i	$-0.258386\pi$
$929$	1278.74			1.37647			0.688234	+	0.725489i	$0.258386\pi$
$930$		0			0
$931$	45.0718	+	326.292i	0.0484122	+	0.350475i
$932$		0			0
$933$	−552.953			−0.592661
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	746.023i		−	0.796182i	−0.917346	−	0.398091i	$-0.869673\pi$
$937$		−	746.023i		−	0.796182i	0.917346	−	0.398091i	$-0.130327\pi$
$938$		0			0
$939$		−	451.599i		−	0.480936i
$940$		0			0
$941$		−	1753.70i		−	1.86365i	−0.362905	−	0.931826i	$-0.618215\pi$
$941$		−	1753.70i		−	1.86365i	0.362905	−	0.931826i	$-0.381785\pi$
$942$		0			0
$943$	−314.227			−0.333220
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1195.95i			1.26288i	0.775425	+	0.631440i	$0.217536\pi$
$947$			1195.95i			1.26288i	−0.775425	+	0.631440i	$0.782464\pi$
$948$		0			0
$949$		−	268.092i		−	0.282500i
$950$		0			0
$951$	11.7603			0.0123662
$952$		0			0
$953$	667.620			0.700546			0.350273	−	0.936648i	$-0.386089\pi$
$953$	667.620			0.700546			0.350273	+	0.936648i	$0.386089\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			250.586i			0.261845i
$958$		0			0
$959$	−806.781			−0.841273
$960$		0			0
$961$	−1083.37			−1.12734
$962$		0			0
$963$	−79.5824			−0.0826401
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1708.02i			1.76630i	0.469086	+	0.883152i	$0.344583\pi$
$967$			1708.02i			1.76630i	−0.469086	+	0.883152i	$0.655417\pi$
$968$		0			0
$969$	1397.05	−	192.979i	1.44174	−	0.199153i
$970$		0			0
$971$		−	216.028i		−	0.222480i	−0.993794	−	0.111240i	$-0.964518\pi$
$971$		−	216.028i		−	0.222480i	0.993794	−	0.111240i	$-0.0354822\pi$
$972$		0			0
$973$		−	889.776i		−	0.914466i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	57.4013			0.0587526			0.0293763	−	0.999568i	$-0.490648\pi$
$977$	57.4013			0.0587526			0.0293763	+	0.999568i	$0.490648\pi$
$978$		0			0
$979$			1762.90i			1.80072i
$980$		0			0
$981$		−	72.1889i		−	0.0735870i
$982$		0			0
$983$	207.431			0.211019			0.105509	−	0.994418i	$-0.466353\pi$
$983$	207.431			0.211019			0.105509	+	0.994418i	$0.466353\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−642.331			−0.650792
$988$		0			0
$989$	−541.780			−0.547805
$990$		0			0
$991$			828.905i			0.836432i	0.908348	+	0.418216i	$0.137345\pi$
$991$			828.905i			0.836432i	−0.908348	+	0.418216i	$0.862655\pi$
$992$		0			0
$993$		−	284.764i		−	0.286771i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1735.54i		−	1.74076i	−0.492381	−	0.870380i	$-0.663873\pi$
$997$		−	1735.54i		−	1.74076i	0.492381	−	0.870380i	$-0.336127\pi$
$998$		0			0
$999$	−697.459			−0.698157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.g.d.949.7		28
5.2	odd	4		1900.3.e.h.1101.11	yes	14
5.3	odd	4		1900.3.e.g.1101.4	✓	14
5.4	even	2	inner	1900.3.g.d.949.22		28
19.18	odd	2	inner	1900.3.g.d.949.21		28
95.18	even	4		1900.3.e.g.1101.11	yes	14
95.37	even	4		1900.3.e.h.1101.4	yes	14
95.94	odd	2	inner	1900.3.g.d.949.8		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.3.e.g.1101.4	✓	14	5.3	odd	4
1900.3.e.g.1101.11	yes	14	95.18	even	4
1900.3.e.h.1101.4	yes	14	95.37	even	4
1900.3.e.h.1101.11	yes	14	5.2	odd	4
1900.3.g.d.949.7		28	1.1	even	1	trivial
1900.3.g.d.949.8		28	95.94	odd	2	inner
1900.3.g.d.949.21		28	19.18	odd	2	inner
1900.3.g.d.949.22		28	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.93221 q^{3} -5.62705i q^{7} -0.402160 q^{9} +O(q^{10})\)	\(q-2.93221 q^{3} -5.62705i q^{7} -0.402160 q^{9} -17.4504 q^{11} -7.35569 q^{13} +25.3144i q^{17} +(2.59985 + 18.8213i) q^{19} +16.4997i q^{21} +19.3107i q^{23} +27.5691 q^{27} +4.89731i q^{29} -45.2147i q^{31} +51.1681 q^{33} -25.2986 q^{37} +21.5684 q^{39} +16.2721i q^{41} +28.0559i q^{43} +38.9299i q^{47} +17.3363 q^{49} -74.2270i q^{51} +81.9687 q^{53} +(-7.62329 - 55.1879i) q^{57} -61.0035i q^{59} -94.1024 q^{61} +2.26297i q^{63} -109.942 q^{67} -56.6231i q^{69} -57.6965i q^{71} +36.4469i q^{73} +98.1940i q^{77} -112.540i q^{79} -77.2188 q^{81} -42.1499i q^{83} -14.3599i q^{87} -101.024i q^{89} +41.3909i q^{91} +132.579i q^{93} -148.954 q^{97} +7.01784 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 104 q^{9}+O(q^{10}) \) 28 * q + 104 * q^9	\( 28 q + 104 q^{9} + 8 q^{11} + 58 q^{19} + 112 q^{39} - 276 q^{49} - 100 q^{61} + 132 q^{81} - 184 q^{99}+O(q^{100}) \) 28 * q + 104 * q^9 + 8 * q^11 + 58 * q^19 + 112 * q^39 - 276 * q^49 - 100 * q^61 + 132 * q^81 - 184 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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